If \(S(y)\)  is a sufficient statistic, then \(p_{\mathrm {ABC}}(\theta \vert y_{\mathrm {obs}})\)  converges to the exact posterior \(p (\theta \vert y_{\mathrm {obs}})\)  as \(h\rightarrow 0\) . Otherwise, \(p_{\mathrm {ABC}}(\theta \vert y_{\mathrm {obs}})\)  converges to the posterior \(p (\theta \vert S(y_{\mathrm {obs}}))\)  as \(h\rightarrow 0\) . There is a gap between \(p (\theta \vert S(y))\)  and \(p (\theta \vert y)\) . We refer to [1]} for a comparative review of choosing low-dimensional summary statistics.

